From sectionally pseudocomplemented posets to pseudoimplicative dual BCK-algebras

A construction is described which, when applicable, associates with a sectionally pseudocomplemented poset a certain operation $$\rightarrow $$ such that $$x \rightarrow y = x'_y$$ whenever $$y \leqslant x$$ . If A is an upper semilattice, then $$x \rightarrow y = (x \hspace{1.111pt}{\vee }\hspace{1.111pt}y)'_y$$ for all x, y. The resulting ordered algebras $$(A,\rightarrow ,1)$$ (recently already considered by the author under the name ‘normal ESP-posets’) are characterized axiomatically: they are particular dual weak BCK-algebras termed pseudoimplicative. In the case when the meet exists in A for every pair of elements bounded below, a normal ESP-poset is a dual BCK-algebra (necessarily pseudoimplicative) if and only if it has distributive upper sections. Moreover, these sections are then Brouwerian semilattices.